1. Field of the Invention
The present invention is related to the field of fuzzy logic and, more particularly, to a system and method for automatically generating a multi-variable fuzzy inference system from given sample sets.
2. Description of the Related Art
It is well known that the fuzzy system is extremely effective in approximating any continuous function on a compact set. The fuzzy system most often provides a powerful alternative to the conventional control system, particularly for complex or ill-defined systems where the system is not easily controllable by a conventional controller. This is so because the fuzzy system is capable of capturing the approximate, qualitative aspects of human knowledge and reasoning needed in the control system. Accordingly, fuzzy systems have recently found an increasingly wider range of applications in industrial applications, household appliances as well as in financial analysis. The applications can include various manufacturing processes, robotics, consumer products such as heat exchangers, warm water pressure control, aircraft flight control, robot control and manipulation, car speed control, power systems and nuclear reactor control, control of a cement kiln, focusing of a camcorder, climate control for buildings, train scheduling, pattern recognition and system modeling, stock trading on a stock exchange and information retrieval, to mention only a few.
It is normally the domain human experts who manually code and refine the fuzzy systems. The performance of a fuzzy system depends critically on the depth of the experts"" understanding of the domain knowledge. Accordingly, the cost of developing a fuzzy system has greatly increased, creating a bottleneck in the application of many complex fuzzy systems in new areas. This has motivated an intensive study of the automatic generation of fuzzy systems, and the subject has often been studied as a data-driven, self-adaptive problem.
In implementing self-adaptive problems by fuzzy systems, one of the biggest problems encountered is what is referred to as a dimensional explosion predicament which is associated with an exponential explosion in the required computational time as the number of variables increases. This is so because the relationship among the multi-variables of the fuzzy rules and membership functions is often quite complicated and more often than not is nonlinear. Because of this, almost all of the existing commercial works available can only deal with a relatively small number of input variables, typically a maximum of five.
In xe2x80x9cA New Approach for the Automatic Generation of Membership Functions and Rules of Multi-Variable Fuzzy Systemxe2x80x9d, (Proceedings of IEEE 1995 International Conference on Neural Networks, Perth, Australia, pp. 1342-1346), Chen et al. presents a new algorithm, PolyNeuFuz, based on polynomial approximation. PolyNeuFuz was later improved to ParNeuFuz, as discussed by Chen et al. in xe2x80x9cA New Scheme for an Automatic Generation for Multi-Variable Fuzzy Systemsxe2x80x9d, to be published in Fuzzy Sets and Systems, 120 (2001), no. 2, pp. 143-149. ParNeuFuz can, in principle, be run in parallel processing if the independence of the input variables is exploited.
Both PolyNeuFuz and ParNeuFuz solve the problem of generating multi-variable fuzzy systems by decomposing the problem to a solution of single input, multiple output fuzzy systems. The ParNeuFuz algorithm exploits a polynomial expansion in approximating the given sample sets, and the time complexity of the algorithms does not depend too heavily on the number of variables used. In principle, both of these algorithms can be applied to generate a fuzzy system with a large number of variables. PolyNeuFuz and ParNeuFuz successfully decompose a self-adaptive problem of multi-variable systems into several steps, and into parallel steps for ParNeuFuz in particular, thus considerably reducing the computational complexity. However, unlike the Fourier series-based approach which is able to provide an accurate error estimate in each of these steps, in decomposition as well as in composition, error estimation or prediction is not easy using the PolyNeuFuz and ParNeuFuz methods.
Accordingly, a need exists for a scheme capable of generating a multi-variable fuzzy inference system that does not encounter the dimensional explosion predicament known in the prior art, while enabling error estimation and prediction.
In view of the foregoing, one object of the present invention is the development of a new Fourier series-based automatic generation scheme of a multi-variable fuzzy inference system that avoids a dimensional explosion predicament.
Another object of the invention is to provide a method that is capable of generating and computing the fuzzy rules and membership functions for each variable independently of the other variables.
A further object of the invention is to provide a Fourier series-based decomposition supporting error analysis that can specify the error bounds or accuracy thresholds at each stage of decomposition and composition to ensure the final precision of the resulting fuzzy system on the original sample set.
In accordance with this and other objects, the present invention presents a novel method of automatically generating a multi-variable fuzzy inference system for any number of practical applications. The method exploits the Fourier series expansion in decomposing the sample sets into simplified tasks of generating a fuzzy system with a single input variable independent of the other variables. The method begins by decomposing a sample set, e.g., xcex9, into an accumulation of a number of set clusters associated with the given input variables, and computing the fuzzy rules and membership functions for each variable, independently of the other variables, by solving a single input multiple outputs fuzzy system extracted from the set cluster. The resulting decomposed fuzzy rules and membership functions are integrated back into the fuzzy system appropriate for the original sample set xcex9, requiring only a moderate computational cost.
The method described in this invention can be used to obtain a stable fuzzy system by retaining any specified accuracy of the resulting multi-variable function on the original sample set. In other words, by taking advantage of the fact that the decomposition is based on a Fourier series, a careful error analysis indicates the relationship between an overall system error and errors at each stage of decomposition and composition so that error bounds or accuracy thresholds of each of these steps may be specified to ensure the final precision of the resulting fuzzy system on the original sample set.
These and other features of the invention, as well as many of the intended advantages thereof, will become more readily apparent when reference is made to the following description taken in conjunction with the accompanying drawings.